PREDICTION OF DIFFERENTIAL JOULE-THOMSON INVERSION CURVES FOR CRYOGENS USING EQUATIONS OF STATE

Nupura Ravishankar1, P.M. Ardhapurkar2, M.D. Atrey3

1Sardar Vallabhbhai National Institute of Technology, Surat – 395 007

2S.S.G.M. College of Engineering, Shegaon – 444 203

3Indian Institute of Technology Bombay, Mumbai – 400 076

In many cryogenic applications, Joule–Thomson (J–T) effect is used to produce low temperatures. The isenthalpic expansion of gas results in lowering of temperature only if the initial temperature of the working fluid is below its characteristic temperature, called the inversion temperature. Inversion curves are helpful in studying the inversion temperatures. The prediction of inversion curves solely depends on the equation of state (EOS) for the working fluid. In the present work, various EOS are explored in order to predict the Joule–Thomson differential inversion curve for various commonly used cryogens viz. nitrogen, argon, carbon dioxide, helium, hydrogen and neon. The widely accepted EOS such as Van der Waals, Redlich–Kwong and Peng–Robinson EOS are used for this purpose.

Key words: Inversion curve, Joule Thomson Effect, Van der Waal, Redlich Kwong, Peng Robinson

INTRODUCTION Joule–Thomson coefficient, µJT and is given as, The Joule-Thomson (J-T) effect has been widely investigated because of its 흏푻 흁푱푻 = ( ) (1) importance for theoretical and practical 흏푷 푯 purposes. It has many cryogenic applications and is widely used in gas The locus of points for which μJT =0 is liquefaction. The effect describes called the differential inversion curve. The the temperature change of a gas or liquid inversion curve divides the p-T plane into when it is forced through a valve or porous two zones as shown in Figure 1. The J–T plug under adiabatic conditions. coefficient is positive inside the inversion The rate of change of temperature with curve, while it is negative outside the curve. respect to pressure in a J-T process (that is, at constant enthalpy) is called the differential Avraham [7], developed a new procedure to investigate the capability of an EOS to predict an experimental inversion curve in a mathematical way. There are many EOS available in the literature which are claimed to be accurate for specific fluids and for a range of operating conditions only. Therefore, selection of suitable EOS to predict accurate inversion curve for the given cryogen is an important step in the simulation or design Figure 1: J-T inversion curve dividing the p-T plane into studies of the cryogenic equipment involving two zones. J–T expansion. The prediction of inversion curves The present work focuses on the depends solely on the equation of state applicability of various EOS for predicting (EOS) that is used for a given working fluid, the J-T differential inversion curve for which operates in a particular p-T region. various commonly used cryogens, Nitrogen This prediction has been studied by various (N2), viz. Argon (Ar) , Carbon dioxide (CO2), researchers [1-2]. Miller [3] suggested that it Helium (He), Neon (Ne) and Hydrogen (H2). would be desirable for constructors of new The paper presents the calculations of EOS to include a comparison of the differential inversion curves and their predicted inversion curve with an comparison with the experimental data experimental one. Prior to 1970, J-T available in the literature. inversion curves were plotted only for a few EOS namely, Van der Waals, Dieterici, THEORY AND ANALYSIS Lennard-Jones-Devonshire, and DeBoer- Michels EOS [4]. Miller calculated inversion The differential inversion curve is defined as curves for the Redlich-Kwong (RK) and the locus of all points where the Joule- Martin EOS, while Juris and Wenzel Thomson coefficient μJT vanishes. It has a calculated inversion curves for the virial, maximum inversion temperature, i.e., the Berthelot, Beattie-Bridgeman, Benedict- maximum temperature when the pressure is Webb-Rubin, Redlich-Kwong, and Martin- zero. For cooling, it is necessary for the gas Hou EOS. to be below this temperature and inside the Dilay and Heldermann [5] used four inversion curve. recently published EOS to calculate the J-T In the present work, three equations of inversion curves; namely, Soave Redlich state, i.e, Van der Waals, RK, PR have been Kwong, Peng Robinson (PR), Perturbed analysed to understand their applicability to hard chain and Lee Kesler EOS. They predict the inversion curves. Eqn. (2) gives concluded that none of the EOS were able the condition for differential inversion curve. to predict the entire inversion curve 휕푉 푇 ( ) − 푉 accurately; however the Lee Kesler EOS 휕푇 휕푇 푃 휇퐽푇 = ( ) = = 0 (2) gave the best overall prediction. The 휕푃 퐻 퐶푝 obtained inversion curves were then compared with the inversion curve developed by Gunn Chueh, and Prausnitz 휕푉 Thus, 푇 ( ) − 푉 = 0 (3) [2] from the data available for simple fluids. 휕푇 푃 Maytal and Shavit [6] developed the integral Which can also be written as, inversion curve concept, involving the locus of all points with a vanishing integral Joule Thomson effect, ∆Th and isothermal 휕푃 휕푃 휇퐽푇 = 푇 ( ) + 푉 ( ) = 0 (4) enthalpy change, ΔHT. Wisniack and 휕푇 푉 휕푉 푇 The condition so derived in Eqn. (4) is In the reduced form it is: applied to the above mentioned EOS based on which the inversion curves are developed 푇푟 Ω푎 for various gases. The following section 푃푟 = − 2 (9) Ω푏(푥−1) Ω √푇푟푥(푥+1) gives these details. 푏

Van der Waals equation of state Where x is the reduced volume and Ωa The Van der Waals EOS is expressed and Ωb are constants with values as given, in Eqn. (5): Ωa =0.4275 푎 (푃 + ) (푉 − 푏) = 푅푇 (5) 푉2 푚 푚 Ωb =0.08664

The Van der Waals constant ‘a’ is a By applying the equation for differential measure of strength of the Van der Waals condition, Tr can be written as, force between the molecules of the gas, while the Van der Waals constant ‘b’ 3 2 ⁄2 Ω푎(5푥+3)(푥−1) 푇푟 = 2 (10) represents effective volume of the gas Ω푏2푥(푥+1) molecules. The equation can be written in its Peng Robinson equation of state reduced form, as given in Eqn. (6), using The PR EOS is as given in Eqn. (11) pressure, temperature and volume at the critical point. The reduced pressure (Pr), 푅푇 푎훼 푃 = − 2 2 (11) temperature (Tr) and volume (Vr) are given 푉−푏 푉 +2푏푉−푏 as P/Pc, V/Vc and T/Tc respectively. Where parameters a, b and α are given 3 1 8푇푟 by: (푃푟 + 2) (푉푟 − ) = (6) 푉푟 3 3 2 2 0.457235푅 푇푐 By applying the equation for differential 푎 = 푝푐 condition, Pr can be written as, 0.077796푅푇푐 [7] 푏 = 푃푟 = 24√3푇푟 − 12푇푟 − 27 푝푐

Redlich Kwong Equation of State 2

The RK EOS [5] is given as: 훼 = (1 + 푘(1 − √푇푟))

푅푇 푎 푃 = − (8) Where k is: 푉−푏 √푇푉(푉+푏) 푘 = 0.37464 + 1.54226휔 − 0.26992휔2 Where parameters ‘a’ and ‘b’ have the same physical significance as in Van der Where ω is the acentric factor, which is Waals EOS. However, their values are a conceptual number useful in the different which are given as below: description of matter. It is a measure of the non sphericity of molecules. 5 2 ⁄2 In the present work, ω has been 0.4275푅 푇푐 푎 = assumed to be zero, which leads to the 푃 푐 value of k=0.37464.

0.08664푅푇 푏 = 푐 푃푐 In the reduced form, the PR EOS is: are used and results are compared with the experimental data [5] for the inversion curve. 2 푇푟 Ω푎[1+푘(1−√푇푟)] As is evident from the figure, the 푃푟 = − 2 2 (12) Ω푏(푥−1) Ω푏(푥 +2푥−1) inversion curves appear to be qualitatively Where x is the reduced volume similar to each other. But when studied closely, they give varied data of the Ωa =0.457235 inversion temperatures. The curves are divided into two temperature zones - a high Ωb =0.077796 temperature branch and a low temperature branch. The low temperature branch of When the equation for differential these curves closely match with the condition is applied, Tr can be written as, experimental curve. However, at higher temperatures, the curves deviate 2 2 2 2 2 푇푟[Ω푎(푥 − 1) (푥 + 1)푘 −Ω푏(푥 + 2푥 − 1) ] significantly from the experimental data. It is + seen in Figure 2, that Van der Waals EOS √푇 [−Ω (푥 − 1)2푘(푘 + 1)(3푥2 + 2푥 + 1)] + gives good prediction of the low temperature 푟 푎 branch of the inversion curve below 2 bar, 2Ω 푥(푥 + 1)(푥 − 1)2(1 + 푘)2 = 0 (13) 푎 whereas RK and PR EOS give good

prediction of the same upto 8 bar. The peak The above derived expressions in pressure also varies for each EOS, being terms of reduced pressure, temperature and less than the experimental data for Van der volume are solved to obtain various values Waals EOS, and more than that for RK and of Pr for different values of Tr to get various PR EOS. It is 9 bar for Van der Waals, 10.8 inversion curves. These reduced forms of bar for RK, 13 for PR and 11.5 bar for the the equations are solved in MATLAB®. For experimental inversion curve. Thus, the high the RK and PR EOS, for different values of temperature branch and the peak pressure reduced volume x, the corresponding values of the inversion curve prove to be sensitive of reduced temperature T are computed r to the EOS that is used. The RK equation of using equations (10) and (13) respectively. state gives the best prediction of the These combinations of x and T are r inversion curve closest to the experimental substituted in the equation of state to get the curve. value of Pr. The cryogenic gases may be classified

into two groups. Group 1, which consists of

N , Ar and CO , has its inversion RESULT AND DISCUSSION 2 2 temperature above the room temperature.

They can be expanded at room temperature The results are plotted in terms of Tr-Pr to obtain cooling effect. Group 2 consists of curves for various gases using different the quantum gases, H , He, Ne, which have EOS. These curves are compared with the 2 inversion temperatures below the room experimentally obtained curve available in temperature. These gases need to be the literature. Based on these curves, T-P precooled below their inversion temperature curves are plotted for each cryogen (gas) for before being expanded, in order to obtain each EOS. Its intersection with the T axis cooling effect. gives the maximum inversion temperature In the present study, the inversion for any cryogen. The maximum inversion curves are developed for cryogens of both temperatures so obtained for various the groups. Figures 3-6 show the inversion cryogens using different EOS are compared curves for gases from group 1 i.e. N , Ar, with the experimental values in the literature. 2 CO respectively, and the Table 1 shows the Figure 2 shows the comparison of 2 comparison of the experimental inversion differential inversion curves obtained from temperatures and those obtained from the various EOS in terms of their reduced developed inversion curves. properties. Van der Waals, PR and RK EOS It may be seen that the Van der Waals value. As seen from Table 1, the difference EOS performs very poorly for these in the predicted and the experimental value cryogens of group 1. The maximum is about 90K for N2, 30K for Ar and 165K for inversion temperature is highly above the CO2. experimental value, while the maximum The most accurate inversion curve is pressure is considerably less than the predicted by using the RK EOS. It predicts experimental values. For N2 and Ar, the the maximum inversion temperature quite deviation in maximum inversion temperature close to the experimental value. The is around 220K, while for CO2, it is more deviation is minimal, i.e, 50K for N2, 10K for than 500K. Ar and 123K for CO2 as seen from Table 1. Figures 5-7 show inversion curves of gases from group 2 namely H2, He, Ne respectively.

Figure 2: Differential Inversion curves

Figure 5: Differential inversion curves for CO2

Figure 3: Differential inversion curves for N2

Figure 6: Differential inversion curves for H2

Figure 4: Differential inversion curves for Ar Figure 7: Differential inversion curves for He

PR EOS does fairly well and is an These quantum gases do not follow the improvement over Van der Waals EOS. As principle of corresponding states. It is due to compared to Van der Waals EOS, the their quantum effects at low temperatures maximum inversion temperature predicted and high pressures. This means, when by PR EOS is closer to the experimental compared at the same reduced temperature and pressure, they do not have the same developed and compared with the compressibility factor. Thus, the quantum experimental values available in literature. It gases do not exhibit ideal gas behaviour. may be seen that RK EOS gives the overall best prediction of the inversion curve. The Van der Waals equation is most inaccurate, while PR EOS results are closer to that of RK. All three equations, namely Van der Waals, PR and RK EOS give good prediction of the lower temperature branch of the inversion curve. Hence, the higher temperature branch and the peak of the inversion curve decide which EOS predicts the best inversion curve. Figure 8: Differential inversion curves for Ne The quantum gases, H , He and Ne, 2 need to be evaluated separately using EOS Table 1: Comparison of maximum inversion temperatures that considers the quantum effects.

Sr. Gas Maximum Inversion Temp (K) No VDW PR RK Actual NOMENCLATURE 1. Nitrogen 854 691 674 621 P- Pressure 2. Argon 1017 825 804 794 T- Temperature 3. Carbon 2053 1666 1623 1500 V- Volume Dioxide Vm- Molar Volume 4. Hydrogen 224 182 177 205 Pr- Reduced pressure 5. Helium 35 28 27 45 Tr- Reduced temperature 6. Neon 300 243 237 250 Vr/x- Reduced volume µJT – Joule Thomson coefficient This is the reason for the variations observed in maximum inversion temperature for the quantum gases. It may be seen that REFERENCES for H2 and He, Van der Waals EOS predicts the best inversion curve, whereas for Ne it is 1. Porter, A. W. Philos. Mag. 1906, 6, 554. PR EOS. It is known that Van der Waals 2. Gunn, R. D.; Chueh, P. L.; Prausnitz, J. EOS is the most inaccurate EOS among the M. Cryogenics 1966, 6 , 324. three EOS studied. Thus, the fact that Van 3. Miller, D. G. Ind. Eng. Chem. Fundam. der Waals EOS predicts the best inversion 1970, 9, 585. curve is contradictory. So, it can be 4. Juris, K.; Wenzel, L. A. AIChE J. 1972, concluded that there is an anomaly in the 18, 684. calculated inversion curves because of the 5. Gary W. Dllay and Robert A. Heldeman, quantum effects. Calculation of Joule thomson inversion Other higher equations of state need to curve from various equations of state, be used for these gases, which take Ind. Eng. Chem. Fundam., Vol 25, No 1, quantum effects of gases into consideration. 1986 6. On the integral Joule Thomson effect- B.-Z. Maytal and A. Shavit, Cryogenics CONCLUSION 1994 Volume 34,Number 1 7. Jaime Wisniak, Hanan Avraham, On The Differential inversion curves from three Joule Thomson effect inversion curve, equations of state are successfully Thermochimica Acta 286 (1996) 33-40